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Self-similar vortex filament motion under the non-local Biot–Savart model

Published online by Cambridge University Press:  10 August 2016

Robert A. Van Gorder
Robert A. Van Gorder*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
*
†Email address for correspondence: [email protected]
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Abstract

One type of thin vortex filament structure that has attracted interest in recent years is that which obeys self-similar scaling. Among various applications, these filaments have been used to model the motion of quantized vortex filaments in superfluid helium after reconnection events. While similarity solutions have been described analytically and numerically using the local induction approximation (LIA), they have not been studied (or even shown to exist) under the non-local Biot–Savart model. In this present paper, we show not only that self-similar vortex filament solutions exist for the non-local Biot–Savart model, but that such solutions are qualitatively similar to their LIA counterparts. This suggests that the various LIA solutions found previously should be valid physically (at least in the small amplitude regime), since they agree well with the more accurate Biot–Savart model.

JFM classification

Mathematical Foundations: Computational methods Mathematical Foundations: Mathematical Foundations Vortex Flows: Vortex dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 802 , 10 September 2016 , pp. 760 - 774
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Copyright
© 2016 Cambridge University Press 

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References

Adachi, H., Fujiyama, S. & Tsubota, M. 2010 Steady-state counterflow quantum turbulence: simulation of vortex filaments using the full Biot–Savart law. Phys. Rev. B 81 (10), 104511.Google Scholar
del Álamo, J., Jimenez, J., Zandonade, P. & Moser, R. D. 2006 Self-similar vortex clusters in the turbulent logarithmic region. J. Fluid Mech. 561, 329–358.Google Scholar
Arms, R. J. & Hama, F. R. 1965 Localized-induction concept on a curved vortex and motion of an elliptic vortex ring. Phys. Fluids 8, 553–559.Google Scholar
Bekarevich, I. L. & Khalatnikov, I. M. 1961 Phenomenological derivation of the equations of vortex motion in He II. Sov. Phys. JETP 13, 643.Google Scholar
Bewley, G. P., Paoletti, M. S., Sreenivasan, K. R. & Lathrop, D. P. 2008 Characterization of reconnecting vortices in superfluid helium. Proc. Natl Acad. Sci. 105 (37), 13707–13710.Google Scholar
Boersma, J. & Wood, D. H. 1999 On the self-induced motion of a helical vortex. J. Fluid Mech. 384, 263–279.Google Scholar
Boffetta, G., Celani, A., Dezzani, D., Laurie, J. & Nazarenko, S. 2009 Modeling Kelvin wave cascades in superfluid helium. J. Low Temp. Phys. 156, 193.Google Scholar
Da Rios, L. S. 1906 Sul moto d’un liquido indefinite con un filetto vorticoso di forma qualunque. Rend. Circ. Mat. Palermo 22, 117.Google Scholar
Das, C., Kida, S. & Goto, S. 2001 Overall self-similar decay of two-dimensional turbulence. J. Phys. Soc. Japan 70 (4), 966–976.Google Scholar
Fernandez, V. M., Zabusky, N. J. & Gryanik, V. M. 1995 Vortex intensification and collapse of the Lissajous-elliptic ring: single-and multi-filament Biot–Savart simulations and visiometrics. J. Fluid Mech. 299, 289–331.Google Scholar
Fonda, E., Meichle, D. P., Ouellette, N. T., Hormoz, S. & Lathrop, D. P. 2014 Direct observation of Kelvin waves excited by quantized vortex reconnection. Proc. Natl Acad. Sci. USA 111, 4707.CrossRefGoogle ScholarPubMed
Fonda, E., Meichle, D. P., Ouellette, N. T., Hormoz, S., Sreenivasan, K. R. & Lathrop, D. P.2012 Visualization of Kelvin waves on quantum vortices. arXiv:1210.5194.Google Scholar
Gutiérrez, S., Rivas, J. & Vega, L. 2003 Formation of singularities and self-similar vortex motion under the localized induction approximation. Commun. Part. Diff. Equ. 28, 927–968.Google Scholar
Hall, H. E. & Vinen, W. F. 1956a The rotation of liquid helium II. I. Experiments on the propagation of second sound in uniformly rotating helium II. Proc. R. Soc. Lond. A 238, 204.Google Scholar
Hall, H. E. & Vinen, W. F. 1956b The rotation of liquid helium II. II. The theory of mutual friction in uniformly rotating helium II. Proc. R. Soc. Lond. A 238, 215.Google Scholar
Kimura, Y. 1987 Similarity solutions of two-dimensional point vortices. J. Phys. Soc. Japan 56, 2024–2030.Google Scholar
Kimura, Y. 2009 Self-similar collapse of a 3D straight vortex filament model. Geophys. Astrophys. Fluid. Dyn. 103, 135–142.Google Scholar
Kimura, Y. 2010 Self-similar collapse of 2D and 3D vortex filament models. Theoret. Comput. Fluid Dyn. 24 (1–4), 389–394.Google Scholar
Kursa, M., Bajer, K. & Lipniacki, T. 2011 Cascade of vortex loops initiated by a single reconnection of quantum vortices. Phys. Rev. B 83 (1), 014515.Google Scholar
Lipniacki, T. 2000 Evolution of quantum vortices following reconnection. Eur. J. Mech. (B/Fluids) 19 (3), 361–378.Google Scholar
Lipniacki, T. 2003a Quasi-static solutions for quantum vortex motion under the localized induction approximation. J. Fluid Mech. 477, 321–337.Google Scholar
Lipniacki, T. 2003b Shape-preserving solutions for quantum vortex motion under localized induction approximation. Phys. Fluids 15 (6), 1381–1395.Google Scholar
Moin, P., Leonard, A. & Kim, J. 1986 Evolution of a curved vortex filament into a vortex ring. Phys. Fluids 29 (4), 955–963.Google Scholar
Pelz, R. B. 1997 Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Phys. Rev. E 55 (2), 1617.Google Scholar
Ricca, R. L. 1991 Rediscovery of Da Rios equations. Nature 352, 561.Google Scholar
Ricca, R. L. 1994 The effect of torsion on the motion of a helical vortex filament. J. Fluid Mech. 273, 241–259.Google Scholar
Ricca, R. L., Samuels, D. C. & Barenghi, C. F. 1999 Evolution of vortex knots. J. Fluid Mech. 391, 29–44.CrossRefGoogle Scholar
Schwarz, K. W. 1985 Three-dimensional vortex dynamics in superfluid 4He: line-line and line-boundary interactions. Phys. Rev. B 31, 5782.Google Scholar
Siggia, E. D. 1985 Collapse and amplification of a vortex filament. Phys. Fluids 28 (3), 794–805.CrossRefGoogle Scholar
Van Gorder, R. A. 2013 Self-similar vortex dynamics in superfluid 4He under the Cartesian representation of the Hall–Vinen model including superfluid friction. Phys. Fluids 25 (9), 095105.Google Scholar
Van Gorder, R. A. 2014 Decay of helical Kelvin waves on a quantum vortex filament. Phys. Fluids 26, 075101.Google Scholar
Van Gorder, R. A. 2015a The Biot–Savart description of Kelvin waves on a quantum vortex filament in the presence of mutual friction and a driving fluid. Proc. R. Soc. Lond. A 471, 20150149.Google Scholar
Van Gorder, R. A. 2015b Helical vortex filament motion under the non-local Biot–Savart model. J. Fluid Mech. 762, 141–155.Google Scholar
Van Gorder, R. A. 2015c Non-local dynamics governing the self-induced motion of a planar vortex filament. Phys. Fluids 27, 065105.CrossRefGoogle Scholar
Van Gorder, R. A. 2015d Translation of waves along quantum vortex filaments in the low-temperature two-dimensional local induction approximation. Phys. Fluids 27, 095104.Google Scholar
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54, 641–663.Google Scholar
Yoshimoto, H. & Goto, S. 2007 Self-similar clustering of inertial particles in homogeneous turbulence. J. Fluid Mech. 577, 275–286.Google Scholar